Identifier
Values
[[]] => [.,.] => [1] => 1
[[],[]] => [.,[.,.]] => [2,1] => 0
[[[]]] => [[.,.],.] => [1,2] => 2
[[],[],[]] => [.,[.,[.,.]]] => [3,2,1] => 0
[[],[[]]] => [.,[[.,.],.]] => [2,3,1] => 0
[[[]],[]] => [[.,.],[.,.]] => [3,1,2] => 0
[[[],[]]] => [[.,[.,.]],.] => [2,1,3] => 1
[[[[]]]] => [[[.,.],.],.] => [1,2,3] => 3
[[],[],[],[]] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 0
[[],[],[[]]] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 0
[[],[[]],[]] => [.,[[.,.],[.,.]]] => [4,2,3,1] => 0
[[],[[],[]]] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 0
[[],[[[]]]] => [.,[[[.,.],.],.]] => [2,3,4,1] => 0
[[[]],[],[]] => [[.,.],[.,[.,.]]] => [4,3,1,2] => 0
[[[]],[[]]] => [[.,.],[[.,.],.]] => [3,4,1,2] => 0
[[[],[]],[]] => [[.,[.,.]],[.,.]] => [4,2,1,3] => 0
[[[[]]],[]] => [[[.,.],.],[.,.]] => [4,1,2,3] => 0
[[[],[],[]]] => [[.,[.,[.,.]]],.] => [3,2,1,4] => 1
[[[],[[]]]] => [[.,[[.,.],.]],.] => [2,3,1,4] => 1
[[[[]],[]]] => [[[.,.],[.,.]],.] => [3,1,2,4] => 1
[[[[],[]]]] => [[[.,[.,.]],.],.] => [2,1,3,4] => 2
[[[[[]]]]] => [[[[.,.],.],.],.] => [1,2,3,4] => 4
[[],[],[],[],[]] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 0
[[],[],[],[[]]] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 0
[[],[],[[]],[]] => [.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 0
[[],[],[[],[]]] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 0
[[],[],[[[]]]] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 0
[[],[[]],[],[]] => [.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 0
[[],[[]],[[]]] => [.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 0
[[],[[],[]],[]] => [.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 0
[[],[[[]]],[]] => [.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 0
[[],[[],[],[]]] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 0
[[],[[],[[]]]] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 0
[[],[[[]],[]]] => [.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 0
[[],[[[],[]]]] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 0
[[],[[[[]]]]] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 0
[[[]],[],[],[]] => [[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 0
[[[]],[],[[]]] => [[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 0
[[[]],[[]],[]] => [[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 0
[[[]],[[],[]]] => [[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 0
[[[]],[[[]]]] => [[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 0
[[[],[]],[],[]] => [[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 0
[[[[]]],[],[]] => [[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 0
[[[],[]],[[]]] => [[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 0
[[[[]]],[[]]] => [[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 0
[[[],[],[]],[]] => [[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 0
[[[],[[]]],[]] => [[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 0
[[[[]],[]],[]] => [[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 0
[[[[],[]]],[]] => [[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 0
[[[[[]]]],[]] => [[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 0
[[[],[],[],[]]] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 1
[[[],[],[[]]]] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 1
[[[],[[]],[]]] => [[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 1
[[[],[[],[]]]] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 1
[[[],[[[]]]]] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 1
[[[[]],[],[]]] => [[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 1
[[[[]],[[]]]] => [[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 1
[[[[],[]],[]]] => [[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 1
[[[[[]]],[]]] => [[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 1
[[[[],[],[]]]] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 2
[[[[],[[]]]]] => [[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 2
[[[[[]],[]]]] => [[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 2
[[[[[],[]]]]] => [[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 3
[[[[[[]]]]]] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 5
[[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 0
[[],[],[],[],[[]]] => [.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 0
[[],[],[],[[]],[]] => [.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => 0
[[],[],[],[[],[]]] => [.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 0
[[],[],[],[[[]]]] => [.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 0
[[],[],[[]],[],[]] => [.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => 0
[[],[],[[]],[[]]] => [.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => 0
[[],[],[[],[]],[]] => [.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => 0
[[],[],[[[]]],[]] => [.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => 0
[[],[],[[],[],[]]] => [.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 0
[[],[],[[],[[]]]] => [.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 0
[[],[],[[[]],[]]] => [.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => 0
[[],[],[[[],[]]]] => [.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 0
[[],[],[[[[]]]]] => [.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 0
[[],[[]],[],[],[]] => [.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 0
[[],[[]],[],[[]]] => [.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => 0
[[],[[]],[[]],[]] => [.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => 0
[[],[[]],[[],[]]] => [.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => 0
[[],[[]],[[[]]]] => [.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => 0
[[],[[],[]],[],[]] => [.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 0
[[],[[[]]],[],[]] => [.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 0
[[],[[],[]],[[]]] => [.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => 0
[[],[[[]]],[[]]] => [.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => 0
[[],[[],[],[]],[]] => [.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 0
[[],[[],[[]]],[]] => [.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 0
[[],[[[]],[]],[]] => [.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 0
[[],[[[],[]]],[]] => [.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => 0
[[],[[[[]]]],[]] => [.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 0
[[],[[],[],[],[]]] => [.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 0
[[],[[],[],[[]]]] => [.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 0
[[],[[],[[]],[]]] => [.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 0
[[],[[],[[],[]]]] => [.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 0
[[],[[],[[[]]]]] => [.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 0
[[],[[[]],[],[]]] => [.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 0
[[],[[[]],[[]]]] => [.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 0
[[],[[[],[]],[]]] => [.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => 0
[[],[[[[]]],[]]] => [.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 0
>>> Load all 196 entries. <<<
[[],[[[],[],[]]]] => [.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 0
[[],[[[],[[]]]]] => [.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 0
[[],[[[[]],[]]]] => [.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => 0
[[],[[[[],[]]]]] => [.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 0
[[],[[[[[]]]]]] => [.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 0
[[[]],[],[],[],[]] => [[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => 0
[[[]],[],[],[[]]] => [[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => 0
[[[]],[],[[]],[]] => [[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => 0
[[[]],[],[[],[]]] => [[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => 0
[[[]],[],[[[]]]] => [[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => 0
[[[]],[[]],[],[]] => [[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => 0
[[[]],[[]],[[]]] => [[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => 0
[[[]],[[],[]],[]] => [[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => 0
[[[]],[[[]]],[]] => [[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => 0
[[[]],[[],[],[]]] => [[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => 0
[[[]],[[],[[]]]] => [[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => 0
[[[]],[[[]],[]]] => [[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => 0
[[[]],[[[],[]]]] => [[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => 0
[[[]],[[[[]]]]] => [[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => 0
[[[],[]],[],[],[]] => [[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => 0
[[[[]]],[],[],[]] => [[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => 0
[[[],[]],[],[[]]] => [[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => 0
[[[[]]],[],[[]]] => [[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => 0
[[[],[]],[[]],[]] => [[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => 0
[[[[]]],[[]],[]] => [[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => 0
[[[],[]],[[],[]]] => [[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => 0
[[[],[]],[[[]]]] => [[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => 0
[[[[]]],[[],[]]] => [[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => 0
[[[[]]],[[[]]]] => [[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => 0
[[[],[],[]],[],[]] => [[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => 0
[[[],[[]]],[],[]] => [[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => 0
[[[[]],[]],[],[]] => [[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => 0
[[[[],[]]],[],[]] => [[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => 0
[[[[[]]]],[],[]] => [[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => 0
[[[],[],[]],[[]]] => [[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => 0
[[[],[[]]],[[]]] => [[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => 0
[[[[]],[]],[[]]] => [[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => 0
[[[[],[]]],[[]]] => [[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => 0
[[[[[]]]],[[]]] => [[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => 0
[[[],[],[],[]],[]] => [[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => 0
[[[],[],[[]]],[]] => [[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => 0
[[[],[[]],[]],[]] => [[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => 0
[[[],[[],[]]],[]] => [[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => 0
[[[],[[[]]]],[]] => [[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => 0
[[[[]],[],[]],[]] => [[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => 0
[[[[]],[[]]],[]] => [[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => 0
[[[[],[]],[]],[]] => [[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => 0
[[[[[]]],[]],[]] => [[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => 0
[[[[],[],[]]],[]] => [[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => 0
[[[[],[[]]]],[]] => [[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => 0
[[[[[]],[]]],[]] => [[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => 0
[[[[[],[]]]],[]] => [[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => 0
[[[[[[]]]]],[]] => [[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => 0
[[[],[],[],[],[]]] => [[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 1
[[[],[],[],[[]]]] => [[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 1
[[[],[],[[]],[]]] => [[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => 1
[[[],[],[[],[]]]] => [[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 1
[[[],[],[[[]]]]] => [[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 1
[[[],[[]],[],[]]] => [[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => 1
[[[],[[]],[[]]]] => [[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => 1
[[[],[[],[]],[]]] => [[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => 1
[[[],[[[]]],[]]] => [[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => 1
[[[],[[],[],[]]]] => [[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 1
[[[],[[],[[]]]]] => [[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 1
[[[],[[[]],[]]]] => [[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => 1
[[[],[[[],[]]]]] => [[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 1
[[[],[[[[]]]]]] => [[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 1
[[[[]],[],[],[]]] => [[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => 1
[[[[]],[],[[]]]] => [[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => 1
[[[[]],[[]],[]]] => [[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => 1
[[[[]],[[],[]]]] => [[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => 1
[[[[]],[[[]]]]] => [[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => 1
[[[[],[]],[],[]]] => [[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => 1
[[[[[]]],[],[]]] => [[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => 1
[[[[],[]],[[]]]] => [[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => 1
[[[[[]]],[[]]]] => [[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => 1
[[[[],[],[]],[]]] => [[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => 1
[[[[],[[]]],[]]] => [[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => 1
[[[[[]],[]],[]]] => [[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => 1
[[[[[],[]]],[]]] => [[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => 1
[[[[[[]]]],[]]] => [[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => 1
[[[[],[],[],[]]]] => [[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 2
[[[[],[],[[]]]]] => [[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 2
[[[[],[[]],[]]]] => [[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => 2
[[[[],[[],[]]]]] => [[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 2
[[[[],[[[]]]]]] => [[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 2
[[[[[]],[],[]]]] => [[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => 2
[[[[[]],[[]]]]] => [[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => 2
[[[[[],[]],[]]]] => [[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => 2
[[[[[[]]],[]]]] => [[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => 2
[[[[[],[],[]]]]] => [[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 3
[[[[[],[[]]]]]] => [[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 3
[[[[[[]],[]]]]] => [[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => 3
[[[[[[],[]]]]]] => [[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 4
[[[[[[[]]]]]]] => [[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 6
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
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Description
The number of strong fixed points of a permutation.
$i$ is called a strong fixed point of $\pi$ if
1. $j < i$ implies $\pi_j < \pi_i$, and
2. $j > i$ implies $\pi_j > \pi_i$
This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3].
The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see St000314The number of left-to-right-maxima of a permutation..
Map
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.
Map
to binary tree: right brother = right child
Description
Return a binary tree of size $n-1$ (where $n$ is the size of an ordered tree $t$) obtained from $t$ by the following recursive rule:
- if $x$ is the right brother of $y$ in $t$, then $x$ becomes the right child of $y$;
- if $x$ is the first child of $y$ in $t$, then $x$ becomes the left child of $y$,
and removing the root of $t$.